On a new fractional-order Logistic model with feedback control

In this paper, we formulate and analyze a new fractional-order Logistic model with feedback control, which is different from a recognized mathematical model proposed in our very recent work. Asymptotic stability of the proposed model and its numerical solutions are studied rigorously. By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function, we show that a unique positive equilibrium point of the new model is asymptotically stable. As an important consequence of this, we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability. Furthermore, we construct unconditionally positive nonstandard finite difference(NSFD) schemes for the proposed model using the Mickens' methodology. It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model. Finally, we report some numerical examples to support and illustrate the theoretical results. The results indicate that there is a good agreement between the theoretical results and numerical ones.     

On relative stability of linear stationary feedback control systems with time delay

A method is presented whereby the absolute and relative stabilityof linear control systems containing transport lag can be determined.As a result feedback systems with variable time delay and loopgain, may be investigated in straightforward manner.     

A model of feedback control system on network and its stability analysis

A model of feedback control system on network is constructed. Some criterions ensuring the robust stability and global exponential stability for the model are established, by applying Lyapunov method and generalized Kirchhoff’s matrix tree theorem. These criterions have a close relation to the topology property of networks. Finally, two examples and simulations are provided to illustrate the theoretical results.     

Dynamic analysis of a turbidostat model with the feedback control

In this paper, a mathematical model with impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of positive order one periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentration of microorganism and substrate. By investigating the periodic solution, the period and the initial point of the periodic solution are given. The results show that turbidostat with impulsive state feedback control tends to an order one periodic solution.     

On a mutualism model with feedback controls

We propose and study a mutualism model with feedback controls. By applying a new differential inequality, we show that the conditions which ensure the permanence of the system are the same as that of the model without feedback controls, which means that the feedback control variables have no influence on the persistent property of the system. Our results not only improve but also complement some existing ones.     

Exponential stability of a kind of wave equation with boundary feedback control

The problem of exponential stability of a kind of wave equation with damping and boundary output feedback control is investigated. The spectral structure of the system operator is analyzed and it is shown that the c0-semigroup generated by the system operator is exponential stable if only the coefficients viscous damping and boundary feedback control are not zeros simultaneously.     

On the control of solutions of viscoelastic equations with boundary feedback

In this paper we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary. For a wider class of relaxation functions and without imposing any restrictive growth assumption on the damping term, we establish an explicit and general decay rate result.     

Boundary feedback control of elastic beam equation with structural damping and stability

In this paper, we consider the partial differential equation of an elastic beam with structural damping by boundary feedback control. First, we prove this closed system is well-posed; then we establish the exponential stability for this elastic system by using a theorem whichbelongs to F. L. Huang^[2]; finally, we discuss the distribution and multiplicity of the spectrum of this system. These results are very important and useful in practical applications.     

Confidence domain in the stochastic competition chemostat model with feedback control

This paper studies a stochastically forced chemostat model with feedback control in which two organisms compete for a single growth-limiting substrate. In the deterministic counterpart, previous researches show that the coexistence of two competing organisms may be achieved as a stable positive equilibrium or a stable positive periodic solution by different feedback schedules. In the stochastic case, based on the stochastic sensitivity function technique,we construct the confidence domains for different feedback schedules which allow us to find the configurational arrangements of the stochastic attractors and analyze the dispersion of the random states of the stochastic model.     

On a stabilizing feedback attitude control

The attitude control of a rotating satellite with two control jets leads to a system of four controlled ordinary differential equations of the form (S) $$dx/dt = X(x) + u_1 Y^1 (x) + u_2 Y^2 (x),x(0) = 0.$$ Our goal is to derive feedback controlsu ₁,u ₂ which automatically stabilize the system (S), i.e., drive the solution to the (uncontrolled) rest solution zero. Let $$(ad^0 X,Y) = Y,(adX,Y) = [X,Y],$$ the Lie product of the vector fieldsX, Y, and inductively $$(ad^{k + 1} X,Y) = [X,(ad^k X,Y)].$$ It is known that, if $$dim span\left\{ {\left( {ad^j X,Y^1 } \right)\left( 0 \right),j = 0,1,...} \right\} = 4,$$ then all points in some neighborhood of zero can be controlled to zero with just the controlu ₁, i.e.,u ₂≡0. In this problem,Y ¹(0), ..., (ad ³ X, Y ¹)(0) are linearly independent. We give a formula for generating the directions (ad ⁱ X, Y ⁱ )(0) as endpoints of admissible trajectories. Our modified feedback control is then formed as follows. Given an ε>0, if the state of system (S) is measured to beq ¹ ∈ ?⁴, we write $$q^1 = \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0),$$ and choose a controlu(t,q ¹) on the interval 0≤t≤ε to drive the solution in the direction $$ - \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0).$$ Thus, we assume that the state is measured (say) at time intervals 0, ε, 2ε, ..., while the control depends on the measured state, but then is open loop during a time interval ε until a new state is measured; hence, the terminologymodified feedback control. Numerical results are included for both the case of one control component and the case of two control components.     

Matching conditions for stability analysis of nonlinear feedback control systems

The differential geometry approach for exact input-output linearizable nonlinear systems usually results in a complicated nonlinear controller that is difficult to implement. To overcome this difficulty, we propose to approximate the nonlinear controller by a canonical piecewise linear expression within an allowable error bound. This design procedure can reduce a tremendous amount of computation in the design and the synthesis. The resulting controller turns out to be fairly simple in general and can achieve many performance specifications. Some sufficient conditions for guaranteeing the closed-loop system stability using this controller design method is derived in the present paper. An application to a chemical reactor system is also briefly discussed.     

On the stability of a population model with nonlocal dispersal

This paper is concerned with a nonlocal dispersal population model with spatial competition and aggregation. We establish the existence and uniqueness of positive solutions by the method of coupled upper-lower solutions. We obtain the global stability of the stationary solutions.     

Delayed feedback control of a chemical chaotic model

The Willamowski–Rössler model system is investigated. It has been found that the system can be locked in a special district: stable without oscillation, periodic-1 oscillation, periodic-2 oscillation by the time delayed feedback. Numerical simulation result has also shown that the initial condition can affect the result of chaos controlling.     

Resonance,stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

This paper examines dynamical behavior of a nonlinear oscillator with a symmetric potential that models a quarter-car forced by the road profile. The primary, superharmonic and subharmonic resonances of a harmonically excited nonlinear quarter-car model with linear time delayed active control are investigated. The method of multiple scales is utilized to obtain first order approximation of response. We focus on the influence of delay in the system. This naturally gives rise to a delay deferential equation (DDE) model of the system. The effect of time delay and feedback gains of the steady state responses of primary, superharmonic and subharmonic resonances are investigated. By means of Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. We describe a method to identify the critical forcing function and time delay above which the system becomes unstable. It is found that proper selection of time-delay shows optimum dynamical behavior. The accuracy of the method is obtained from the fractal basin boundaries.     

On the stability of a wet-land model

Mathematical models for studying the effect of ecological changes caused by the excessive growth of wild changes on the existence of various species in wet-land are investigated. Local stability criteria is also discussed.